Difference between revisions of "Partial derivative"
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===Divergence=== | ===Divergence=== | ||
| + | When <math>\mathbf{f}</math> is a [[vector]]-valued function or vector field whose values are in the same dimensions as its arguments, the [[Vector#Dot(Scalar) Product|dot product]] of <math>\nabla</math> with <math>\mathbf{f}</math> yields the '''divergence''' <cmath>\nabla \cdot \mathbf{f} = \frac{\partial \mathbf{f}_{x_1}}{\partial x_1} + \frac{\partial \mathbf{f}_{x_2}}{\partial x_2} + \dots + \frac{\partial \mathbf{f}_{x_n}}{\partial x_n},</cmath> | ||
| + | where the notation <math>\mathbf{f}_{x_i}</math> refers to the <math>x_i</math>-direction component of vector <math>\mathbf{f}</math>. | ||
| + | |||
| + | Informally, the divergence measures how much the field "spreads out" from a point. This is because the term in the sum corresponding to each dimension is positive if the change in the component of the field in the dimension agrees with change in the the argument coordinate in the dimension (pushing the field away from the point) and negative if the change in the component disagrees with change in the coordinate (pulling the field toward the point), with magnitude determined by the rate of such change of the field. | ||
===Curl=== | ===Curl=== | ||
Revision as of 14:10, 3 May 2022
A partial derivative of a function of many variables is the derivative of that function with respect to one of its arguments.
For example, if 
then 
has three partial derivatives at the point 
:
Del operator
The del operator, or nabla symbol, written 
, represents the vector ![\[\left( \frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2}, \dots , \frac{\partial}{\partial x_n} \right)\]](http://latex.artofproblemsolving.com/8/6/e/86e44828adbbf7ec1e319339fa7a1b9a3fafbb2a.png)
where the value 
is the arity (number of arguments) of the function in question.
Gradient
The product of and a function 
is ![\[\nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots , \frac{\partial f}{\partial x_n} \right),\]](http://latex.artofproblemsolving.com/f/d/7/fd744045e9b83ca619907644c18532c9f41adcf5.png)
a vector storing, in order, all of the partial derivatives of .
The gradient applies when is a scalar-valued function of many variables. For example, the gradient of temperature in a closed room is 
, where 
, 
, and 
are the Cartesian coordinates in the three spatial dimensions: length, width, and height, respectively. 
and 
are likely to be close to zero at most points, but 
probably has a small positive value, since the air nearer the ceiling (greater ) is warmer than the air nearer the floor (lesser ). Therefore, the typical direction of the gradient vector 
is close to upwards.
Divergence
When 
is a vector-valued function or vector field whose values are in the same dimensions as its arguments, the dot product of with yields the divergence ![\[\nabla \cdot \mathbf{f} = \frac{\partial \mathbf{f}_{x_1}}{\partial x_1} + \frac{\partial \mathbf{f}_{x_2}}{\partial x_2} + \dots + \frac{\partial \mathbf{f}_{x_n}}{\partial x_n},\]](http://latex.artofproblemsolving.com/4/0/5/405ad060e1c099259d82eb026c1f6652b6bfb45b.png)
where the notation 
refers to the 
-direction component of vector .
Informally, the divergence measures how much the field "spreads out" from a point. This is because the term in the sum corresponding to each dimension is positive if the change in the component of the field in the dimension agrees with change in the the argument coordinate in the dimension (pushing the field away from the point) and negative if the change in the component disagrees with change in the coordinate (pulling the field toward the point), with magnitude determined by the rate of such change of the field.
